1. CEM in action Computed surface currents on prototype military aircraft at 100MHz The plane wave is incident from left to right at nose on incidence. The currents re-radiate back to the source radar (and so can be detected)

2. 83 Camaro at 1 GHz Irradiation of a 83 Camaro at 1 GHz by a Hertzian dipole.

3. Inlet Scattering SimulationMeasurement > 2,000,000 unknowns

4. Corrugated Horn Antenna

5. Microstrip Antenna Array Current distribution Radiation patterns

6. Time Varying Current Distribution

7. EMP Microwave pulse penetrating a missile radome containing a horn antenna. Wave is from right to left at 15° from boresight.

8. Broadband Analysis of Wave Interactions with Nonlinear Electronic Circuitry 25 cm 5 cm 17.5 cm 10 cm 1 cm 20 cm 4.5 cm 6 cm x y z 0.5 cm 15 cm 1 cm y 500  Voltages on the varistors Voltage (kV) EM solvers permit analysis of wave broadband EMC/EMI phenomena, and the assessment of electronic upset and terrorism scenarios

9. Scattering at 3 GHz from Full Fighter Plane (fast solvers) Bistatic RCS of VFY218 at 3 GHz 8 processors of SGI Origin 2000 # of Unknowns N = 2 millions FIES LUD CG Memory Matrix-fill LUD One-RHS (GB) (days) (years) (hrs) 5 0.1 9 32,000 600.0 200 4 32,000 600.0 500 AZ

10. Computational Electromagnetics computational electromagnetics High frequency rigorous methods IE DE MoM FDTD TLM field based current based GO/GTD PO/PTD TDFD TDFD VM FEM

11. Computational Electromagnetics Electromagnetic problems are mostly described by three methods: Differential Equations (DE)  Finite difference (FD, FDTD) Integral Equations (IE)  Method of Moments (MoM) Minimization of a functional (VM)  Finite Element (FEM) Theoretical effort less more Computational effort more less

13. Fields Fields: A space (and time) varying quantity –Static field: space varying only –Time varying field: space and time varying –Scalar field: Magnitude varies in space (and time) –Vector field: Magnitude & direction varies in space (and time) Moving Fields…... Electromagnetic waves

14. Time Harmonic Fields Fields that vary periodically (sinusoidally) with time Time Harmonic Scalar Fields Phasor Transform P Real, time harmonic scalar Complex Number (Phasor)

15. Maxwell’s Equations in Differential Form Faraday’s Law Ampere’s Law Gauss’s Law Gauss’s Magnetic Law

17. Faraday’s Law S C

18. Ampere’s Law

19. Gauss’s Law

20. Gauss’s Magnetic Law “all the flow of B entering the volume V must leave the volume” (no magnetic charges!)

24. CONSTITUTIVE RELATIONS  r  o = permittivity (F/m)  o = 8.854 x 10 -12 (F/m)  r  o = permeability (H/m)  o = 4  x 10 -7 (H/m)  = conductivity (S/m)

25. POWER and ENERGY Stored magnetic power (W) Stored electric power (W) Supplied power (W) Dissipated power (W) What is this term?

26. POWER and ENERGY Stored magnetic power (W) Stored electric power (W) Supplied power (W) Dissipated power (W) What is this term? P s = power exiting the volume through radiation W/m 2 Poynting vector

27. TIME HARMONIC EM FIELDS Assume all sources have a sinusoidal time dependence and all materials properties are linear. Since Maxwell’s equations are linear all electric and magnetic fields must also have the same sinusoidal time dependence. They can be written for the electric field as: is a complex function of space (phasor) called the time-harmonic electric field. All field values and sources can be represented by their time-harmonic form. Euler’s Formula

28. PROPERTIES OF TIME HARMONIC FIELDS Time derivative: Time integration:

29. TIME HARMONIC MAXWELL’S EQUATIONS Employing the derivative property results in the following set of equations:

30. TIME HARMONIC EM FIELDS BOUNDARY CONDITIONS AND CONSTITUTIVE PROPERTIES The constitutive properties and boundary conditions are very similar for the time harmonic form: Constitutive Properties General Boundary Conditions PEC Boundary Conditions

31. TIME HARMONIC EM FIELDS IMPEDANCE BOUNDARY CONDITIONS If one of the material at an interface is a good conductor but of finite conductivity it is useful to define an impedance boundary condition:  1,      2,      1 >>  2

32. POWER and ENERGY: TIME HARMONIC Time average magneticenergy (J) Time average electric energy (J) Supplied complex power (W) Dissipated real power (W) Time average exiting power

33. CONTINUITY OF CURRENT LAW vector identity time harmonic

34. SUMMARY Frequency Domain Time Domain

35. Wave Equation Vector Identity Time Dependent Homogenous Wave Equation (E-Field)

36. Wave Equation Source-Free Time Dependent Homogenous Wave Equation (E-Field) Source Free Source-Free Lossless Time Dependent Homogenous Wave Equation (E-Field) Lossless

37. Wave Equation Source-Free Time Dependent Homogenous Wave Equation (H-Field) Source Free Source Free and Lossless

38. Wave Equation: Time Harmonic Source Free Lossless Time DomainFrequency Domain Source Free Lossless “Helmholtz Equation”

39. MOST POPULAR COMPUTATIONAL ELECTROMAGNETICS ALGORITHMS FINITE DIFFERENCE (FD) METHODS Example: Finite difference time domain (FDTD) INTEGRAL EQUATION METHODS (IE) Example: Method of Moments (MoM) VARIATIONAL METHODS Example: Finite element method (FEM)

40. Numerical Differentiation “FINITE DIFFERENCES”

41. Introduction to differentiation Conventional Calculus –The operation of diff. of a function is a well-defined procedure –The operations highly depend on the form of the function involved –Many different types of rules are needed for different functions –For some complex function it can be very difficult to find closed form solutions Numerical differentiation –Is a technique for approximating the derivative of functions by employing only arithmetic operations (e.g., addition, subtraction, multiplication, and division) –Commonly known as “finite differences”

42. Taylor Series Problem: For a smooth function f(x), Given: Values of f(x i ) and its derivatives at x i Find out: Value of f(x) in terms of f(x i ), f(x i ), f  (x i ), …. x y f(x) f(x i ) xixi

43. Taylor’s Theorem If the function f and its n+1 derivatives are continuous on an interval containing x i and x, then the value of the function f at x is given by

44. Finite Difference Approximations of the First Derivative using the Taylor Series (forward difference) x y f(x) f(x i ) xixi x i+1 f(x i+1 ) h Assume we can expand a function f(x) into a Taylor Series about the point x i+1 h

45. Finite Difference Approximations of the First Derivative using the Taylor Series (forward difference) Assume we can expand a function f(x) into a Taylor Series about the point x i+1 Ignore all of these terms

46. Finite Difference Approximations of the First Derivative using the Taylor Series (forward difference) x y f(x) f(x i ) xixi x i+1 f(x i+1 ) h

47. Finite Difference Approximations of the First Derivative using the forward difference: What is the error? The first term we ignored is of power h 1. This is defined as first order accurate. First forward difference

48. Finite Difference Approximations of the First Derivative using the Taylor Series (backward difference) x y f(x) f(x i-1 ) x i-1 xixi f(x i ) h Assume we can expand a function f(x) into a Taylor Series about the point x i-1 -h

49. Finite Difference Approximations of the First Derivative using the Taylor Series (backward difference) Ignore all of these terms First backward difference

50. Finite Difference Approximations of the First Derivative using the Taylor Series (backward difference) x y f(x) f(x i-1 ) x i-1 xixi f(x i ) h

51. Finite Difference Approximations of the Second Derivative using the Taylor Series (forward difference) y x f(x) f(x i ) xixi x i+1 f(x i+1 ) h x i+2 f(x i+2 ) (1) (2) (2)-2* (1)

52. Finite Difference Approximations of the Second Derivative using the Taylor Series (forward difference) y x f(x) f(x i ) xixi x i+1 f(x i+1 ) h x i+2 f(x i+2 ) Recursive formula for any order derivative

53. Higher Order Finite Difference Approximations

54. Centered Difference Approximation (1) (2) (1)-(2)

55. Finite Difference Approximations of the First Derivative using the Taylor Series (central difference) x y f(x) f(x i-1 ) x i-1 xixi f(x i ) h x i+1 f(x i+1 )

56. Second Derivative Centered Difference Approximation (central difference) (1) (2) (1)+(2)

57. Using Taylor Series Expansions we found the following finite-differences equations FORWARD DIFFERENCE BACKWARD DIFFERENCE CENTRAL DIFFERENCE

58. Forward finite-difference formulas

59. Centered finite difference formulas

60. Finite Difference Approx. Partial Derivatives Problem: Given a function u(x,y) of two independent variables how do we determine the derivative numerically (or more precisely PARTIAL DERIVATIVES) of u(x,y)

61. Pretty much the same way STEP #1: Discretize (or sample) U(x,y) on a 2D grid of evenly spaced points in the x-y plane

62. x axis y axis xixi x i+1 x i-1 x i+2 yjyj y j+ 1 y j-1 y j-2 u(x i,y j )u(x i+1,y j ) u(x i,y j-1 ) u(x i,y j+1 ) u(x i-1,y j ) u(x i-1,y j+1 ) u(x i-1,y j-1 ) u(x i-1,y j-2 )u(x i,y j-2 ) u(x i+1,y j-1 ) u(x i+1,y j-2 ) u(x i+1,y j+1 ) u(x i+2,y j ) u(x i+2,y j-1 ) u(x i+2,y j-2 ) u(x i+2,y j+1 ) 2D GRID

63. x axis y axis ii+1i-1 i+2 j j+1 j-1 j-2 u i,j u i+1,j u i-1,j u i,j-1 u i,j+1 SHORT HAND NOTATION

64. Partial First Derivatives Problem: FIND recall:

65. Partial First Derivatives Problem: FIND xx yy These are central difference formulas Are these the only formulas we could use? Could we use forward or backward difference formulas?

66. Partial First Derivatives: short hand notation Problem: FIND xx yy

67. Partial Second Derivatives Problem: FIND recall:

68. Partial Second Derivatives Problem: FIND xx yy

69. Partial Second Derivatives: short hand notation Problem: FIND xx yy

70. FINITE DIFFERENCE ELECTROSTATICS Electrostatics deals with voltages and charges that do no vary as a function of time. Poisson’s equation Laplace’s equation Where,  is the electrical potential (voltage),  is the charge density and  is the permittivity.

71. oo 11 22 33 FINITE DIFFERENCE ELECTROSTATICS: Example Find  (x,y) inside the box due to the voltages applied to its boundary. Then find the electric field strength in the box.

72. Electrostatic Example using FD Problem: FIND xx yy

73. Electrostatic Example using FD Problem: FIND If  x =  y

74. Electrostatic Example using FD Problem: FIND Iterative solution technique: (1)Discretize domain into a grid of points (2)Set boundary values to the fixed boundary values (3)Set all interior nodes to some initial value (guess at it!) (4)Solve the FD equation at all interior nodes (5)Go back to step #4 until the solution stops changing (6)DONE

75. Electrostatic Example using FD MATLAB CODE EXAMPLE